1. False. For example, carbon dioxide (which has four lone pairs, two on each oxygen) is a nonpolar molecule.

2. This depends what you mean by symmetrical. The standard non-technical definition of "symmetrical" is having a mirror plane, but obviously this cannot be correct because water is mirror-plane-symmetrical and is highly polar. If a molecule has an inversion center* it cannot be polar.

3. False. For instance boron trifluoride (BF3) which is planar and has three fluorine atoms surrounding the central boron atom, is nonpolar.

*An inversion center is a point at the center of a molecule (or any other geometric form, abstract or concrete) for which every atom in the molecule has a partner atom that is the same distance from the center, but in exactly the opposite direction. If there are any atoms without a matching atoms on the other side of the center of the molecule, then the molecule does not have an inversion center (unless the unmatched atom is at the center). (https://en.wikipedia.org/wiki/Molecular_symmetry#Elements)

Any molecule without an inversion center must be polar (even if its dipole moment is so close to 0 that it is considered "non-polar," like pentane)

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